2.1. Microjetting in liquid–liquid flow focusing

This paper analyses the effect of a soluble surfactant on the stability of the liquid–liquid hydrodynamic focusing configuration sketched in figure 1, commonly referred to as flow focusing. In this axisymmetric configuration, a liquid of density $\rho _i$ and viscosity $\mu _i$ is injected at a constant flow rate $Q_i$ across a cylindrical feeding capillary of radius $\hat {R}_c$. This liquid flows surrounded by an outer liquid current of density $\rho _o$ and viscosity $\mu _o$, immiscible with the former. The outer current is injected at a constant flow rate $Q_o$ across a converging nozzle concentric with the inner one. The inner and outer phases coflow through the nozzle, which ends in a tube with diameter $\hat {D}$. The surface tension without surfactant is $\hat {\gamma }_*$. As explained below, we will use this value as the characteristic surface tension $\hat {\gamma }_c$.

Figure 1.

Sketch of the fluid configuration.

Under certain parameter conditions, the system adopts the microjetting mode of tip streaming (Montanero & Gañán-Calvo Reference Montanero and Gañán-Calvo2020). In this mode, a jet with a diameter much smaller than $2\hat {R}_c$ and $\hat {D}$ is emitted from the tip of the fluid meniscus attached to the feeding capillary. The jet and the outer stream coflow through the discharge tube (figure 1). The jet is expected to break up into droplets beyond the fluid domain considered in our analysis. The stability of microjetting in a similar flow focusing configuration was studied by Cabezas et al. (Reference Cabezas, Rubio, Rebollo-Muñoz, Herrada and Montanero2021) both experimentally and numerically. The results of the global stability analysis were in good agreement with the experiments.

The geometry considered in the present work is the same as that analysed by Cabezas et al. (Reference Cabezas, Rubio, Rebollo-Muñoz, Herrada and Montanero2021) except that it incorporates a discharge cylindrical tube (marked with a red line in figure 1) not considered in that work. In this way, the inner and outer streams do not slow down beyond the nozzle neck. Perturbations growing in the tube are convected downstream. This eliminates the unstable modes associated with the jet (absolute instability) (Huerre & Monkewitz Reference Huerre and Monkewitz1990) and allows one to focus on the meniscus stability. It is known that tip streaming instability originates in the tapering meniscus in most relevant realizations (Montanero & Gañán-Calvo Reference Montanero and Gañán-Calvo2020). Therefore, the stability limit for the present configuration is not expected to differ significantly from that of the geometrical configuration analysed by Cabezas et al. (Reference Cabezas, Rubio, Rebollo-Muñoz, Herrada and Montanero2021). As shown in § 4, the flow fully develops near the tube entrance. For this reason, the spectrum of eigenvalues characterizing the linear dynamics becomes practically insensitive to the tube length for relatively small values of this parameter.

The geometry studied in this work can be regarded as a hybrid between flow focusing and confined selective withdrawal (He et al. Reference He, Campo-Cortés, Goral, López-León and Gordillo2019), where the focusing effect is produced by a cylindrical tube located in front of the feeding capillary. There are, however, noticeable differences between these two configurations. The shape of the focusing element (either an orifice or a tube) significantly affects the microjetting stability (López et al. Reference López, Cabezas, Montanero and Herrada2022).

2.2. The surfactant

This paper examines the effect of a surfactant dissolved in the inner phase on the microjetting stability. The surfactant transport across the bulk is described in terms of the diffusion coefficient $\hat {{\mathcal {D}}}_i$. In particular, the diffusion flux normal to the interface is

(2.1)\begin{equation} \hat{{\mathcal{J}}}_{\mathcal{D}}={-}\hat{{\mathcal{D}}}_i \frac{\partial \hat{c}}{\partial n}, \end{equation}

where $\hat {c}$ is the bulk surfactant concentration, $n$ is the direction normal to the interface and $\partial \hat {c}/\partial n$ is evaluated at the interface (the interface sublayer). The diffuse layer also depends on the ionic strength of ionic surfactants, such as sodium dodecyl sulphate (SDS). This effect is not considered in our analysis.

We adopt the kinetic model for the adsorption/desorption flux

(2.2a–c)\begin{equation} \hat{{\mathcal{J}}}= \hat{{\mathcal{J}}}_a-\hat{{\mathcal{J}}}_d,\quad \hat{{\mathcal{J}}}_a=\hat{k}_a\hat{c}_s\left(1- \frac{\hat{\varGamma}}{\hat{\varGamma}_{\infty}}\right),\quad \hat{{\mathcal{J}}}_d=\hat{k}_d \hat{\varGamma}, \end{equation}

where $\hat {{\mathcal {J}}}$ is the net sorption flux, $\hat {{\mathcal {J}}}_a$ and $\hat {{\mathcal {J}}}_d$ are the adsorption and desorption fluxes, respectively, $\hat {k}_a$ and $\hat {k}_d$ are the adsorption and desorption constants, respectively, $\hat {c}_s$ is the bulk surfactant concentration $\hat {c}$ evaluated at the interface, $\hat {\varGamma }$ is the surfactant surface concentration (the surface coverage, measured in mols per unit area) and $\hat {\varGamma }_{\infty }$ is the maximum packing density.

The surfactant molecules absorbed on the interface are transported by convection and diffusion. The surfactant surface diffusion is determined by the surface diffusion coefficient $\hat {{\mathcal {D}}}_{s}$, which typically takes values similar to those of the bulk diffusion coefficient $\hat {{\mathcal {D}}}_i$ (Tricot Reference Tricot1997).

We assume that the transfer of surfactant between the sublayer next to the interface and the interface occurs only in the form of monomers. This implies that $\hat {c}$ is the volumetric concentration of monomers even if the surfactant concentration exceeds the critical micelle concentration $\hat {c}_{cmc}$ and, therefore, there are micelles in the inner liquid. Since the micelles do not contribute to $\hat {c}_s$, values of $\hat {c}_s$ exceeding the critical micelle concentration correspond to higher experimental values of the surfactant concentration. We will consider surfactant concentrations larger than $\hat {c}_{cmc}$. However, these concentrations will be sufficiently small for the dynamical effects of the micelles to be negligible.

The surfactant is convected throughout the incompressible inner liquid phase. For this reason, the surfactant concentration in the feeding capillary is approximately the same as that in the reservoir, $\hat {c}_\infty$. The bulk diffusion coefficient $\hat {{\mathcal {D}}}_i$ corresponds to very high values of the Péclet number in most experiments. This implies that the bulk surfactant concentration is almost uniform in the inner phase, except within a thin diffusive layer of thickness $\hat {\lambda }_D$ next to the interface.

As shown in § 4, $\hat {\lambda }_D/\hat {R}_c\sim 10^{-3}$ for the values of $\hat {{\mathcal {D}}}_i$, $\hat {k}_a$ and $\hat {R}_c$ of the reference case studied in §§ 4–6. The disparity between the thickness of the diffusive layer next to the interface and the relevant dimensions of the problem poses a major challenge to the numerical simulation. This challenge can be circumvented by assuming the approximation $\hat {c}_s\sim \hat {c}_{\infty }$, which leads to accurate results under certain conditions (Lytra et al. Reference Lytra, Vlachomitrou and Pelekasis2024). As explained in § 3, we resolve the surfactant mass boundary layer by using a spatial discretization based on Chebyshev spectral collocation points (Khorrami Reference Khorrami1989).

The effect of the surface concentration $\hat {\varGamma }$ on the surface tension $\hat {\gamma }$ is described by the Langmuir equation of state (Tricot Reference Tricot1997)

(2.3)\begin{equation} \hat{\gamma}=\hat{\gamma}_*+\hat{\varGamma}_{\infty} \hat{R}_g T\ln \left(1-\frac{\hat{\varGamma}}{\hat{\varGamma}_{\infty}}\right), \end{equation}

where $\hat {R}_g$ is the gas constant and $T$ is the temperature. This equation can be derived from (2.2a–c) and the Gibbs isotherm (Chang & Franses Reference Chang and Franses1995). The surface tension variation over the interface gives rise to the local soluto-capillarity effect and Marangoni stress.

As mentioned in the Introduction, we do not consider surface rheology effects. The effects of the shear $\mu _1^S$ and dilatational $\mu _2^S$ surface viscosities can be quantified in terms of the superficial Ohnesorge numbers $Oh_{1,2}^S=\mu _{1,2}^S(\rho _i \hat {\gamma }_c \hat {R}_c^3)^{-1/2}$, which measure the relative importance of the surface viscous stresses and the capillary pressure. Ponce-Torres et al. (Reference Ponce-Torres, Rubio, Herrada, Eggers and Montanero2020) concluded that the SDS viscosities are at most of the order of $10^{-7}\ {\rm Pa} \,{\cdot }\, {\rm s} \,{\cdot }\, {\rm m}$. Then, the Ohnesorge numbers are at most of the order of $10^{-2}$ in our problem, which indicates that SDS surface viscosities can be neglected.

2.3. Dimensionless numbers

We choose the feeding capillary radius $\hat {R}_c$, the visco-capillary velocity $v_{\gamma \mu }=\hat {\gamma }_c/\mu _i$ and the capillary pressure $\hat {\gamma }_c/\hat {R}_c$ as the characteristic quantities, where $\hat {\gamma }_c$ is a characteristic value of the surface tension, whose choice will be discussed below.

In the absence of surfactants, the problem can be formulated in terms of the density and viscosity ratios, $\rho =\rho _o/\rho _i$ and $\mu =\mu _o/\mu _i$, the Ohnesorge and capillary numbers

(2.4a,b)\begin{equation} {Oh}_i=\left(\frac{\mu_i}{\rho_i v_{\gamma\mu} \hat{R}_c}\right)^{1/2}=\frac{\mu_i}{(\rho_i \hat{\gamma}_c \hat{R}_c)^{1/2}} \quad \text{and} \quad \textit{Ca}=\frac{\mu_o U}{\hat{\gamma}_c}, \end{equation}

and the flow rate ratio $Q=Q_i/Q_o$. Here, $U=4Q_o/({\rm \pi} \hat {D}^2)$ is the mean velocity in the nozzle neck. The capillary number is the ratio of the characteristic viscous stress $\mu _o U/\hat {R}_c$ to the capillary pressure $\hat {\gamma }_c/\hat {R}_c$.

When a soluble surfactant is added to the disperse phase, the following dimensionless numbers are considered as well: the bulk and surface Péclet numbers ${Pe}=\hat {R}_cv_{\gamma \mu }/\hat {{\mathcal {D}}}_i$ and ${Pe}_s=\hat {R}_cv_{\gamma \mu }/\hat {{\mathcal {D}}}_s$, the dimensionless adsorption and desorption constants $k_a=\hat {k}_a \hat {c}_{cmc} \hat {R}_c/(\hat {\varGamma }_{\infty }v_{\gamma \mu })$ and $k_d=\hat {k}_d \hat {R}_c/v_{\gamma \mu }$, the Marangoni (elasticity) number $Ma=\hat {\varGamma }_{\infty } R_g T/\hat {\gamma }_c$ and the reservoir surfactant concentration $c_\infty =\hat {c}_\infty /\hat {c}_{cmc}$.

There are two natural choices for the characteristic surface tension: (i) the value $\hat {\gamma }_*$ corresponding to the surfactant-free case and (ii) the equilibrium value $\hat {\gamma }_{eq}$ corresponding to reservoir concentration $\hat {c}_\infty$. As shown in § 4, surfactant molecules are convected downstream by the interface, which results in values of the meniscus surface tension much larger than $\hat {\gamma }_{eq}$. For this reason, we consider $\hat {\gamma }_*$ in the definition of the dimensionless numbers.

For a fixed geometry, the set of dimensionless numbers governing the problem is $\{\rho$, $\mu$, Oh$_i$, ${\textit {Ca}}$, $Q$; $Pe, Pe_s$, $k_a$, $k_d$, $Ma$, $c_\infty \}$. In a typical experimental run, the microfluidic device, the fluids and the surfactant are fixed. For this reason, the variables $Ca$ and $Q$ can be regarded as the control parameters. To determine the stability limit, one selects the outer flow rate $Q_o$ and progressively decreases the inner flow rate $Q_i$ until its minimum value $Q_{i\,{min}}$ for microjetting is reached. The experiment is usually repeated for several values of $Q_o$. Therefore, the goal is to determine the dimensionless minimum flow rate $Q_{min}=Q_{i\,{min}}/Q_o$ as a function of ${\textit {Ca}}$. In the presence of a surfactant, the function $Q_{min}=Q_{min}(Ca)$ must be obtained for different values of $c_\infty$ (see § 7).

5.1. Surfactant effect on the perturbations and base flow

The surfactant influence on the system stability through both the perturbations and the base flow can be studied by comparing the results with and without surfactant. In other words, we compare the solutions for $c_\infty =0$ and $c_\infty =1.47$. Figure 10 shows the spectrum of eigenvalues in these two cases. The values for $\gamma =1$ (without soluto-capillarity) and $\tau ^{Ma}=0$ (without Marangoni stress) correspond to $c_\infty =0$, while the values for $\gamma \neq 1$ and $\tau ^{Ma}\neq 0$ correspond to $c_\infty =1.47$. The addition of surfactant stabilizes the microjetting mode of flow focusing, significantly reducing the critical growth rate. The eigenfrequencies of the subdominant modes are hardly affected by the surfactant monolayer. This means that the major effect of the surfactant monolayer on the system's linear dynamics occurs precisely through the critical eigenmode. This effect significantly affects the stability limit. The minimum flow rate ratio in the absence of surfactant $Q_{min}=0.0125$ decreases to 0.005 when the surfactant is added.

Figure 10.

Spectrum of eigenvalues around $\tilde {\omega }=0.6$ for $Ca=0.349$ and $Q=0.005$. The stars are the results when neither soluto-capillarity nor Marangoni stress is considered. The triangles are the results obtained only considering soluto-capillarity. The squares are the results obtained only considering Marangoni stress. The circles correspond to the results when the two effects are considered. The arrows indicate the displacement of the critical mode eigenvalue due to soluto-capillarity and Marangoni stress. The values of $Ca$ and $Q$ approximately correspond to marginal stability (${max}\{\omega _i\}=0$) in the presence of surfactant. The eigenvalues have been made dimensionless with the inertio-capillary time $t_{ic}=(\rho _i \hat {R}_c^3/\gamma _*)^{1/2}$.

A supercritical oscillatory Hopf bifurcation ($\omega _r^*\neq 0$) causes the flow instability, as in most tip streaming configurations (Montanero & Gañán-Calvo Reference Montanero and Gañán-Calvo2020). The presence of the surfactant monolayer does not alter this result. The critical perturbation oscillation is linked to the presence of the jet. The jet convects capillary modes, translating into an oscillatory behaviour in the Eulerian frame of reference. As in other tip streaming configurations, $\omega _r^*$ is commensurate with the inverse of the inertio-capillary time based on the meniscus size.

The importance of soluto-capillarity and Marangoni stress has been assessed separately by ‘turning off’ the corresponding terms in the interface boundary conditions (3.6) and (3.10). When both soluto-capillarity ($\gamma \neq 1$) and Marangoni stress ($\tau ^{Ma}\neq 0$) are present, $Q_{min}$ decreases from 0.0125 to 0.005. If the dependence $\gamma (\varGamma )$ is retained in (3.6) but the term $\tau ^{Ma}$ is set to 0 in (3.10), $Q_{min}$ decreases from 0.0125 to 0.0075. This means that the two factors significantly contribute to the microjetting stabilization. As expected, there is a correspondence between the magnitude of the decrease in the minimum flow rate due to those factors and the corresponding decrease in the critical growth rate (figure 10): the larger the decrease in $Q_{min}$ associated with either soluto-capillarity or Marangoni stress, the larger the corresponding decrease in $\omega ^*_i$.

The surfactant monolayer not only affects the minimum flow rate ratio but also influences the shape of the interface deformation caused by the critical eigenmode. Figure 11 compares $\delta F(z)$ for $c_{\infty }=0$ and 1.47 at the corresponding stability limit. The jet diameter is larger without surfactant because the flow rate ratio is larger in that case. The surfactant monolayer increases the interface deformation in the meniscus–jet transition region. This deformation expands upstream and downstream in the absence of surfactant.

Figure 11.

Interface location $F_0(z)$ (a) and perturbation of the interface location, $\delta F(z)$, (b) for $Ca=0.349$ and $c_{\infty }=0$ and 1.47. The results correspond to the minimum flow rate ratios $Q=0.0125$ and 0.005 corresponding to $c_{\infty }=0$ and 1.47, respectively. The values of Re[$\delta F$] are normalized by dividing them by the absolute values of the corresponding minimum values.

5.2. Surfactant effect on the perturbations

In the previous section, we analysed the influence of the surfactant on the microjetting stability through its effect on both the perturbations and the base flow. Now, we exclusively examine the surfactant effect on the perturbations, i.e. for a given base flow. Specifically, we consider the marginally stable base flow for $c_{\infty }=1.47$ and study the perturbations around this base flow with and without the surfactant effect only on the perturbations (not on the base flow).

In the presence of a surfactant monolayer, the perturbation produces a variation $\delta \gamma (z)\, \textrm {e}^{-\textrm {i}\omega t}+\textrm {c.c.}$ of the surface tension with respect to its value $\gamma _0(z)$ in the base flow. This variation is the distinct feature of the perturbations suffered by a surfactant-loaded interface. For this reason, we analyse its effect on the spectrum of eigenfrequencies. The amplitude $\delta \gamma$ can be calculated from the linearization of (3.20) as $\delta \gamma =- {Ma} \delta \varGamma$.

The linearized equation for the balance of normal stresses at the interface reads

(5.3)\begin{equation} {-}\delta p^{(i)}+\delta\tau_n^{(i)}=\gamma_0 \delta\kappa+\delta p_{\gamma}-\delta p^{(o)}+\delta\tau_n^{(o)}, \end{equation}

where $\delta p^{(i)}$, $\delta p^{(o)}$, $\delta \tau _n^{(i)}$ and $\delta \tau _n^{(o)}$ are the perturbations of the hydrostatic pressure and normal viscous stresses, $\gamma _0(z)$ is the surface tension in the base flow and

(5.4)\begin{align} \delta\kappa&=\frac{1}{F_0^2\sqrt{1+F_{0z}^2}}\delta F+\frac{F_0 F_{0z}+F_0 F_{0z}^3-3 F_0^2 F_{0z} F_{0zz}}{F_02(1+F_{0z}^2)^{5/2}} \delta F_z\nonumber\\ &\quad +\frac{F_0^2+F_0^2 F_{0z}^2}{F_0^2(1+F_{0z}^2)^{5/2}} \delta F_{zz}, \end{align}

is the curvature perturbation. The surface tension perturbation $\delta \gamma$ enters into (5.3) through the capillary pressure perturbation $\delta p_{\gamma }=\delta \gamma \kappa _0$, where $\kappa _0$ is the curvature (3.7) in the base flow.

The linearized equation for the balance of tangential stresses is

(5.5)\begin{equation} \delta\tau_t^{(i)}+\delta\tau^{Ma}=\delta\tau_t^{(o)}, \end{equation}

where $\delta \tau _t^{(i)}$ and $\delta \tau _t^{(o)}$ are the perturbations of the tangential viscous stresses, and

(5.6)\begin{equation} \delta\tau^{Ma}={\delta\tau^{Ma}_*}-\delta F_z \frac{\gamma_{0z}F_{0z}}{\sqrt{1+F_{0z}^2}}, \end{equation}

is the Marangoni stress perturbation. The surface tension perturbation $\delta \gamma$ enters into (5.6) through the contribution $\delta \tau ^{Ma}_*=-\delta \gamma _z(1+F_{0z}^{2})^{1/2}$ to the Marangoni stress perturbation.

As mentioned above, the presence of a surfactant monolayer affects the perturbations (excluding the effect on the base flow) through the terms $\delta p_{\gamma }$ and $\delta \tau ^{Ma}_*$ in (5.3) and (5.6), respectively. These terms are proportional to surface tension perturbation $\delta \gamma$, which is the distinct characteristic of a surfactant-covered interface. We analyse the influence of these terms on the microjetting stability in figure 12.

Figure 12.

Spectrum of eigenvalues around $\tilde {\omega }=0.6$ for $Ca=0.349$ and $Q=0.005$. The stars are the results when neither of the terms proportional to $\delta \gamma$ is considered. The triangles are the results obtained only considering $\delta p_{\gamma }$. The squares are the results obtained only considering $\delta \tau ^{Ma}_*$. The circles correspond to the results when the two terms are considered. The eigenvalues have been made dimensionless with the inertio-capillary time $t_{ic}=(\rho _i \hat {R}_c^3/\gamma _*)^{1/2}$.

Firstly, we consider the eigenfrequencies calculated by setting $\delta \tau ^{Ma}_*=0$ to assess the effect of $\delta p_{\gamma }$. As observed, this term hardly alters the eigenfrequencies of the subdominant modes and slightly reduces the growth rate of the critical perturbation. Secondly, we consider the eigenfrequencies for $\delta p_{\gamma }=0$ to determine the effect of $\delta \tau ^{Ma}_*$. The term $\delta \tau ^{Ma}_*$ hardly alters the eigenfrequencies of the subdominant perturbations but considerably decreases the growth rate of the critical eigenmode. Lastly, figure 12 also shows the combined effect of these two terms, which leads to the marginal stability of the base flow.

The decrease in $\omega _i^*$ for $\delta \tau ^{Ma}_*\neq 0$ and $\delta p_{\gamma }\neq 0$ (figure 12) is similar to that produced for $\gamma \neq 1$ and $\tau ^{Ma}\neq 0$ (figure 10), i.e. when the surfactant effect on both the perturbations and the base flow is considered. This suggests that the change in the base flow alone caused by the surfactant plays a secondary role. The main conclusion of the analysis of figure 12 is that the Marangoni stress $\delta \tau ^{Ma}_*$ caused by the surface tension perturbation $\delta \gamma$ is the major stabilizing mechanism in the perturbations.

The results in the Appendix show how the capillary pressure perturbation $\delta p_{\gamma }$ and the Marangoni stress perturbation $\delta \tau ^{Ma}_*$ drive the inner liquid from one of the sides of the perturbation neck towards the neck. This is a stabilizing effect because it opposes the neck thinning.

5.3. Effect of the surfactant on the base flow

We close our stability analysis by studying the influence of the surfactant on the microjetting stability through its effect only on the base flow. To this end, we compare the eigenfrequencies obtained in the absence of surfactant ($c_{\infty }=0$) with those calculated for $c_{\infty }=1.47$ but $\delta p_{\gamma }=\delta \tau ^{Ma}_*=0$. The latter corresponds to the base flow with surfactant but without the surfactant effect on the perturbation.

Figure 13 shows that the surfactant monolayer slightly decreases the frequencies of the subdominant perturbations. The growth (damping) rates remain practically constant. This effect is almost the same as that observed when the surfactant influence on the perturbations and the base flow was considered (figure 10), which confirms that subdominant eigenmodes are only affected through the change of the base flow. The critical mode growth rate decreases when the base flow for $c_{\infty }=1.47$ is considered. However, this decrease is much smaller than that produced by the Marangoni stress $\delta \tau ^{Ma}_*$ associated with the perturbed surface tension $\delta \gamma$ (figure 12).

Figure 13.

Spectrum of eigenvalues around $\tilde {\omega }=0.6$ for $Ca=0.349$ and $Q=0.005$. The squares are the results without surfactant. The diamonds are the results for $\delta p_{\gamma }=\delta \tau ^{Ma}_*=0$ (with surfactant but without its effect on the perturbation). The eigenvalues have been made dimensionless with the inertio-capillary time $t_{ic}=(\rho _i \hat {R}_c^3/\gamma _*)^{1/2}$.

The results presented in § 4 allow us to discuss the relatively minor role played by the base flow in the minimum flow rate instability. This instability can be explained in terms of mass conservation. Suppose all the parameters characterizing the problem are fixed except for the flow rate ratio $Q$ (i.e. the inner flow rate $Q_i$). For sufficiently small values of this parameter, the flow rate dragged by the outer viscous stream becomes practically independent of $Q$. The dragged flow rate must be replaced through the feeding capillary. This sets a minimum value of $Q$ ($Q_i$) below which the steady microjetting regime cannot be sustained. The Marangoni stress collaborates with inner viscous stress to balance the outer viscous stress at the interface (figure 11), reducing the dragged flow rate. This allows one to reduce the value of $Q$ ($Q_i$) while preserving mass conservation.

The stabilizing effect of the surfactant through the base flow can also be explained as follows. As shown in figure 11, the instability originates in the meniscus–jet transition region. The fluid particles experience an adverse pressure gradient ($dp_{\gamma 0}/ds>0$) when moving next to the interface due to the increase in the capillary pressure $p_{\gamma 0}=\gamma _0\kappa _0$ downstream ($s$ is the interface intrinsic coordinate) (figure 14). The resulting force opposing the flow in the meniscus tip constitutes another destabilizing mechanism. The surfactant monolayer considerably reduces the capillary pressure in that region (figure 9) and, consequently, the adverse pressure gradient (figure 14). This effect stabilizes the flow.

Figure 14.

Interface location $F_0(z)$ and surface gradient of the capillary pressure $\textrm {d} p_{\gamma 0}/\textrm {d} s$ along the interface. The results were calculated for $Ca=0.349$, $Q=0.005$ and $c_{\infty }=0$ and 1.47. The dotted vertical lines indicate the position at the symmetry axis of the stagnation points.

7.1. Influence of the surfactant concentration and the adsorption constant

We analyse the influence of the surfactant concentration $c_{\infty }$ and adsorption constant $k_a$ in figure 17. As observed in figure 5, $c_{\infty }=1.47$ and $k_a=4.71\times 10^{-4}$ lead to surfactant surface concentrations relatively close to the maximum packing density in the meniscus tip and the emitted jet. Increasing $c_{\infty }$ and $k_a$ above those values hinders the numerical method convergence and may render the Langmuir equation of state inaccurate. For this reason, we restrict our analysis to $c_{\infty }\leq 1.47$ and $k_a\leq 4.71\times 10^{-4}$.

Figure 17.

Value of $Q_{min}$ as a function of $c_{\infty }$ (a) and $k_a$ (b). The values of the rest of the parameters are those specified in § 4.

As expected, the minimum flow rate ratio decreases as the surfactant concentration and adsorption constant increase (figure 17). We find approximately linear dependencies of $Q_{min}$ on $c_{\infty }$ and $k_a$ within the intervals of those parameters analysed. There is a relatively small effect on the flow stability for, say, $c_{\infty }\lesssim 0.5$ ($\hat {c}_{\infty }\lesssim 0.5\hat {c}_{cmc}$). This agrees with experimental results, which indicate that the surfactant concentration must exceed by far the critical micelle concentration to produce noticeable effects (Moyle et al. Reference Moyle, Walker and Anna2012; López et al. Reference López, Cabezas, Montanero and Herrada2022).

Figure 18 shows the minimum flow rate ratio $Q_{min}$ as a function of $c_{\infty } k_a$. The circles are the results obtained for $k_a=4.71\times 10^{-4}$ and surfactant concentrations $c_{\infty }$ in the interval $[0,1.47]$. The triangles are the results obtained for $c_{\infty }=1.47$ and adsorption constants $k_a$ in the interval $[0,4.71\times 10^{-4}]$. Convection ensures that the volumetric surfactant concentration at the interface approximately equals $c_{\infty }$. Therefore, ${\mathcal {J}}_a\simeq k_a c_{\infty }(1-\varGamma )$, which means that the amount of surfactant absorbed on the interface is directly proportional to $k_a c_{\infty }$. For this reason, the microjetting stability depends on $c_{\infty }$ and $k_a$ through the product $k_a c_{\infty }$ (figure 18).

Figure 18.

Value of $Q_{min}$ as a function of $k_a c_{\infty }$. The circles are the results obtained for $k_a=4.71\times 10^{-4}$ and surfactant concentrations $c_{\infty }$ in the interval $[0,1.47]$. The triangles are the results obtained for $c_{\infty }=1.47$ and adsorption constants $k_a$ in the interval $[0,4.71\times 10^{-4}]$. The values of the rest of the parameters are those specified in § 4.

Figure 19 shows the excellent agreement between the simulation results and the prediction (4.3) for the jet radius $R_j$. The symbols correspond to both stable and unstable base flows with and without surfactant. The surfactant monolayer does not affect the jet radius, which approximately scales as $Q^{1/2}$.

Figure 19.

Jet radius $R_j$ as a function of the flow rate ratio $Q$ for the simulations in figure 18. The symbols are the simulation results, while the solid line is the prediction (4.3).

7.2. Influence of the capillary number

This section closes by analysing the role of the capillary number (figure 20). We do not consider capillary numbers larger than 0.35 because the numerical method fails to converge to the solution in the presence of the surfactant. This probably occurs because the value $\varGamma =1$ is exceeded at some interface point during the simulations. As expected, $Q_{min}$ decreases as $Ca$ increases both with and without surfactant. For a given microfluidic device and liquid–liquid system, the increase in capillary number is produced by an increase in the outer flow rate. One may wonder whether the decrease in the flow rate ratio $Q_{min}$ is only due to the increase of $Q_o$ or, conversely, because $Q_i$ decreases as well. The flow rate ratio $Q_{min}$ can be calculated as a function of the inner flow rate as

(7.1)\begin{equation} Q_{min}=\frac{4\mu_o Q_i}{{\rm \pi} \hat{D}^2 \gamma_*} Ca^{{-}1}. \end{equation}

Figure 20.

Value of $Q_{min}$ as a function of $Ca$ with (solid symbols) and without (open symbols) surfactant. The dashed lines are the isolines of $Q_i$. The values of the rest of the parameters are those specified in § 4.

Figure 20 shows three isolines of $Q_i$ for the values of $\mu _o$, $\hat {D}$ and $\gamma _*$ mentioned § 4. Increasing the capillary number (the outer flow rate) decreases the critical inner flow rate. This behaviour differs from that of gaseous flow focusing, in which the minimum inner flow rate hardly depends on the driving force intensity for sufficiently large values of that quantity (Acero et al. Reference Acero, Ferrera, Montanero and Gañán-Calvo2012).

The interface velocity increases with the capillary number (4.5), which entails the reduction of the residence time

(7.2)\begin{equation} t_r=\int_{z_e}^{L_n-z_e} \frac{(1+F_{0z}^2)^{1/2}}{v_0^s}\, {\rm d} z, \end{equation}

of the interface element in the numerical domain. Figure 21 shows the residence time and the surfactant transported by the jet per unit length, $2{\rm \pi} F_{0e}\varGamma _{0e}$, where $F_{0e}$ and $\varGamma _{0e}$ are the jet radius and surfactant surface concentration evaluated at the numerical domain exit, respectively. The decrease in the residence time reduces the amount of surfactant carried by the jet.

Figure 21.

Residence time $t_r^*$ (open symbols) and surfactant carried by the jet per unit length, $2{\rm \pi} F_{0e}\varGamma _{0e}$, at the numerical domain exit (solid symbols) as a function of $Ca$. The residence time has been made dimensionless using the adsorption characteristic time $\hat {t}_a=\hat {\varGamma }_{\infty }/(\hat {k}_a\hat {c}_{\infty })$; i.e. $t_r^*=t_r (\hat {R}_c\mu _i/\hat {\gamma }_*)/\hat {t}_a$.

As mentioned in § 4, the interface element area drastically decreases in the meniscus tip, which increases the surfactant surface concentration. This effect is more noticeable as $Ca$ increases due to the decrease in $Q_{min}$ (and therefore $R_j$) (figure 20). In fact, $\hat {\varGamma }\simeq \hat {\varGamma }_{\infty }$ ($\varGamma \simeq 1$) at the meniscus tip for $Ca=0.349$ (figure 5) even though the residence time in terms of the adsorption characteristic time, $\hat {t}_a=\hat {\varGamma }_{\infty }/(\hat {k}_a\hat {c}_{\infty })$, is much smaller than unity (figure 21). The competition between the residence time and surface contraction explains the non-monotonic dependency of the jet surface tension (surfactant surface concentration) on the capillary number (figure 22).

Figure 22.

Jet surface tension $\gamma _{0e}$ at the numerical domain exit as a function of $Ca$.

The surfactant stabilizing effect increases with the capillary number. Specifically, the relative reduction of $Q_{min}$ for $c_{\infty }=1.47$ monotonically increases with $Ca$ (figure 20). This contrasts the non-monotonic dependency of the jet surface tension on the capillary number discussed above (figure 22). For $Ca\leq 0.3$, the surfactant stabilizing effect increases with $Ca$ (figure 20) even though the jet surface tension increases with $Ca$ (figure 21). This contradicts the common assumption that adding surfactant favours tip streaming simply because it reduces the meniscus tip surface tension.